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(a) determine the average value of the given function over the given interval, and (b) find the $x$ -coordinate at which the function assumes its average value.$$f(x)=\frac{x}{x^{2}+1},[1,2]$$

(a) $1 / 2 \ln 5 / 2$(b) 1.52846

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 7

Substitution and Properties of Definite Integrals

Integrals

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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(a) determine the average …

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Determine the average valu…

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So the average value function is basically a formula where you do one over the upper bound and it's a lower bound. Then the integral from the two bounds 1 to 2, which they give you the problem of the function. The X uh So what I would do is use u substitution to let the you equally denied. So then d you could equally taken a derivative two x dx and then from there, um, just divide that 1/2 x over. So now I can rewrite this problem, first of all to minus one is just one and one by one is just one. So I really don't need that anymore. Then I need to change my balance because this was in terms of X. I'm gonna change in terms of you. So plug one in for X squared is one plus one is to plug two and four X squared is four plus one is five. I like to leave that X alone, replace this X squared, plus one with you and then replace DX with this 1/2 x d. You based off of this statement right here and then you can see that the excess cancel out and then you can do the anti derivative, which would be that constant stays there. The anti derivative of one over you is natural log with you from 2 to 5. And it's pretty often that my students forget so that one half is just a concept, uh, natural log of five minus natural log of two as you follow your rules of, um, finding the integral, the anti derivative. But then my students tend to forget that subtracting with a log is the same thing as dividing inside. So one half times natural log of five halves is the average value. So what do you have to do next to actually find the X values is set that cancer, this numerical value, this is just a number you have to set that equal to the original problem. X over X squared plus one. Yeah, um, it's only right that X over X squared plus one must equal that numerical value. So what I would do is go to a graphing calculator and type this into why one, um and then type that one half natural log of five halves into y two and again this is just a number. So this will be a horizontal line and find that point of intersection and it might intersect more than once. But we only care about the interval for X. That's bigger than one, but smaller than two based off of what was given to us in the problem. Uh, and that happens at 1.5 to 8. Access is about 1.5 okay to it. You have it, uh, your answer to a your answer to be.

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